[jjk166]

The "i" is there because it is a convenient way in our system of mathematics to write out such an equation, but that really comes from the fact that complex numbers have two dimensionality. Our best understanding of the universe demands that higher dimensionality, not necessarily the imaginary-ness.

Yes a different mathematical formulation may be rewritten into this imaginary form, and thus is mathematically equivalent. But by the same logic a heliocentric system of elliptical orbits is mathematically equivalent to a geocentric system of epicycles. From one perspective there is a certain deeper meaning there - the universe has no absolute reference frame; but if you view your cosmos in terms of epicycles its very difficult to develop an understanding of what drives those epicycles, namely gravity. Likewise thinking about quantum mechanics in terms of of imaginary numbers may allow for accurate calculations, but nevertheless be an intellectual stumbling block for understanding why the universe is this way.

I personally have no issue with "imaginary" numbers having real physical meaning. Our inability to process the square root of negative 1 seems more like a limitation of our ape brains than the universe, and likewise for the majority of quantum weirdness. But in throwing up my hands saying the question can not be answered, I have guaranteed that I will never find the answer even if it does indeed exist.

[ogogmad]

The phase space formulation of QM uses less complex numbers than the Schrodinger one: it models states using quasi-probability distributions, where the "probabilities" behave in all the usual ways except they can go negative. Interestingly, the classical limit of this (that is, when h goes to zero) still has negative probabilities in it.

[Retric]

The issue with epicycles is you need an infinite number of them to produce the actual orbits and with an infinite number of epicycles you can describe any shape. Thus it is as complex as the underlying data.

Quantum Mechanics on the other hand is incredibly constrained and therefore actually says something.

[jjk166]

And pure ellipses as predicted by newtonian gravitation also don't line up with actual orbits perfectly. In both cases they are just models approximating reality, one of which happens to be more elegant. I don't know how anyone would be able to jump straight from epicycles to general relativity.

Quantum mechanics likewise is just an approximation of quantum field theories.

[Retric]

It’s not about elegance for the sake of it. The number of constants in a theory provides a meaningful point of comparison, especially if you need to increase them after an experiment.

[jjk166]

Epicycles wasn't a theory, it was a model. It did not try to explain why the planets moved in the sky as they did, it only predicted where they'd be. Neither, for that matter, were copernican or keplerian mechanics theories. They too required unending tweaking because they also were only approximations of what was actually happening. For the first few centuries after heliocentrism was proposed, it gave worse results, and demanded more tweaking. What really won people over was that the phases of the moons of jupiter were accurately predicted by the model as well. The only way to achieve that result with epicycles was to rearrange everything to be mathematically equivalent to a heliocentric model.

You can reconstruct our modern understanding of the motion of the planets in the reference frame of a static earth and produce a mathematically equivalent path that draws out epicycles which predict the positions of planets with exactly the same accuracy as our regular formulations. You can rework the representation of the laws of gravity such that they spit out positions in this reference frame. It is an equally valid model of the cosmos, with exactly the same number of starting assumptions, it's just remarkably more complex.

[Retric]

It started from an actual theory based around the assumption that spherical motion was perfect. They needed 2 which did actually work for a while, eventually the most accurate model needed ~17 with people giving up on the underlying theory as the number of terms destroyed the initial idea.

Today with vastly more data and more accurate measurements you’d need effectively infinite terms, which makes it more obvious but you don’t need that level of absurdity to render judgment.

[jjk166]

No it didn't. Epicycles were from the get go nothing but an attempt to fit a mathematical function to observed data to predict future positions of planets. It's a geometric method of curve fitting which is a weaker form of the fourier series, and the system was developed by greek mathematicians trying to improve upon Babylonian computations that didn't even have a geometric model. There is a reason that the moon, the only thing in the cosmos that does in fact orbit the earth, has the most complicated series of epicycles to describe its motion.

Ptolemy rejects Aristotle's cosmology which relied on perfect spherical motion. Ptolemy really did believe that the planets moved according to his model (ie it wasn't just a pure computational tool) but he was very clear that his model was based purely on mathematics. Not only did he not give a reason for why the cosmos should take this form, he openly speculates that the answer is unknowable, and works under the assumption "maybe they can move wherever they want and they just like moving this way."

Further, cycles were not added over time [1]. On day one there were 31 cycles and circles, and these were exactly the same ones being used at the time of Copernicus. You also don't need many epicycles to accurately produce a path identical to keplerian orbits. Completely arbitrary orbits can be described with finite epicycles. [2] Indeed the problem was that Ptolemy didn't fit the data by adding more epicycles, but instead through the Equant, which moved the positions of the centers of the epicycles, which meant adding more epicycles would not make it more accurate. The story of ever more epicycles being added to a bloated old theory that was streamlined by heliocentrism is a modern myth.

[1] https://diagonalargument.com/2025/05/20/from-kepler-to-ptole...

[2] https://web.math.princeton.edu/~eprywes/F22FRS/hanson_epicyc...

[IcyWindows]

More complete astronomy data from telescopes showed that epicycles needed to be even more complicated then they were.

If we manage to find better tools for QM where we don't need to perform as much post-selection of experimental data, perhaps we'll also find a simpler model.